To Do List
period_half.The object sessDat has data from all 6 sessions.
0 is the settings player.loc and player.price the subject was initialized at.21 is the player.price the subject would be at if the period continued.Variables in sessDat
session.codeparticipant.code is a unique subject identifyer.player.period_numberplayer.subperiod_numberperiod_half either “First Half” or “Second Half”. NA if period 0 or 21.player.loc locationplayer.price priceplayer.boundary_lo and player.boundary_hi are the high and low boundary for this player currentlygroup_size number of players in the groupgroup_size_str a string for the group sizeplayer.transport_cost shopping cost, 0.10, 0.25, 0.40, 0.60player.mc mill cost, 0.05, 0.15, 0.25player.rp reserve price, 0.8, 0.9, 1.0score_subperiod this player’s current scorescore_total currency period’s total score for this player.Summary of sessions and subjects.
| Number of Players | Sessions | Subjects | Periods Per Session |
|---|---|---|---|
| Four Player | 3 | 72 | 15 |
| Two Player | 3 | 48 | 15 |
Sessions were run at the New York University Abu Dhabi and the United Arab Emirates with undergraduate students between Oct 17 and Oct 19th, 2017.
Subjects earned on average $84.25 from the experiment. After a 30 AED show-up fee and rounding up to the 5 AED, subjects walked away on average with $114.25
The experiment was conducted with oTree (Citation: Chen, D.L., Schonger, M., Wickens, C., 2016. oTree - An open-source platform for laboratory, online and field experiments. Journal of Behavioral and Experimental Finance, vol 9: 88-97) subjects were recruited with hroot (Citation: Bock, Olaf, Ingmar Baetge & Andreas Nicklisch (2014). hroot – Hamburg registration and organization online tool. European Economic Review 71, 117-120)
Hypothesis 1. Static mark-ups will be lower in more competitive (higher N) markets.
In the plot below,
In the pilot we had a spread of transport costs from 0.1 to 1.0. Between 0.1 and 0.5 there wasn’t a huge difference in price, only at 0.75 and 1.0 did we see a substantial increase in markups. In this design we only had a spread of transport costs between 0.1 and 0.6, and we don’t see a consistent increase in price as transport costs increase.
In the plot below,
Comparing prices in both treatments. - We see with greater competition there are lower prices accross all shopping costs.
| playerNum | 0.1 | 0.25 | 0.4 | 0.6 |
|---|---|---|---|---|
| Four Player | 0.31 (±0.0123) | 0.23 (±0.0176) | 0.28 (±0.0117) | 0.31 (±0.024) |
| Two Player | 0.55 (±0.0263) | 0.41 (±0.039) | 0.5 (±0.0255) | 0.44 (±0.0419) |
Now, looking just at the later half of each period, subperiods 11 to 20, (remove from final)
| playerNum | 0.1 | 0.25 | 0.4 | 0.6 |
|---|---|---|---|---|
| Four Player | 0.31 (±0.0123) | 0.23 (±0.0176) | 0.28 (±0.0117) | 0.31 (±0.024) |
| Two Player | 0.55 (±0.0263) | 0.41 (±0.039) | 0.5 (±0.0255) | 0.44 (±0.0419) |
Strong evidence for Hypothesis 1.
Looking at the average prices within a period (all 20 subperiods) with the same player number and transport cost, there is a statistically significant difference between prices at each transport level between player number treatments.
Even comparing t = 6.0 in the four player game – the transport cost in which the four-player game with highest prices – to t = 0.25 in the two player game – in which prices were the lowest in the two-player game – the two player game has statistically significantly higher prices (p-value < 0.001).
##
## Welch Two Sample t-test
##
## data: player.price[(playerNum == "Two Player" & player.transport_cost == and player.price[(playerNum == "Four Player" & player.transport_cost == 0.25)] and 0.6)]
## t = 5.5709, df = 176.38, p-value = 9.331e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.06173099 0.12946116
## sample estimates:
## mean of x mean of y
## 0.4129291 0.3173331
##
## Wilcoxon rank sum test with continuity correction
##
## data: player.price[(playerNum == "Two Player" & player.transport_cost == and player.price[(playerNum == "Four Player" & player.transport_cost == 0.25)] and 0.6)]
## W = 12880, p-value = 1.696e-07
## alternative hypothesis: true location shift is not equal to 0
Hypothesis 2 - There is a positive relationship between shopping costs and mark-ups.
| Shopping Cost | N | Mean Price | Median Price | Standard Error | |
|---|---|---|---|---|---|
| Four Player | 0.10 | 648 | 0.312 | 0.303 | 0.005 |
| Four Player | 0.25 | 168 | 0.229 | 0.195 | 0.009 |
| Four Player | 0.40 | 576 | 0.280 | 0.260 | 0.004 |
| Four Player | 0.60 | 168 | 0.311 | 0.299 | 0.008 |
| Two Player | 0.10 | 432 | 0.547 | 0.523 | 0.008 |
| Two Player | 0.25 | 112 | 0.412 | 0.404 | 0.015 |
| Two Player | 0.40 | 384 | 0.500 | 0.482 | 0.008 |
| Two Player | 0.60 | 112 | 0.443 | 0.436 | 0.012 |
Recall there were 72 subjects in the four-player treatment and 48 subjects in the two-player treatment.
First, within the two player game, comparing prices in t = 0.1 and t = 0.6 (see below), there is to be a statistically significant difference.
There is a relationship between prices and shopping cost treatments. In higher shopping cost settings subjects tended to have higher prices.
##
## Welch Two Sample t-test
##
## data: mean_price[player.transport_cost == 0.1] and mean_price[player.transport_cost == 0.6]
## t = 7.0137, df = 218.42, p-value = 2.867e-11
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.0748557 0.1333672
## sample estimates:
## mean of x mean of y
## 0.5472454 0.4431339
##
## Wilcoxon rank sum test with continuity correction
##
## data: mean_price[player.transport_cost == 0.1] and mean_price[player.transport_cost == 0.6]
## W = 33222, p-value = 1.12e-09
## alternative hypothesis: true location shift is not equal to 0
In the four-player game the relationship, at least between the lowest and highest shopping cost, does not appear stronger.
##
## Welch Two Sample t-test
##
## data: mean_price[player.transport_cost == 0.1] and mean_price[player.transport_cost == 0.6]
## t = 0.06797, df = 310.47, p-value = 0.9459
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.01850285 0.01982693
## sample estimates:
## mean of x mean of y
## 0.3118495 0.3111875
##
## Wilcoxon rank sum test with continuity correction
##
## data: mean_price[player.transport_cost == 0.1] and mean_price[player.transport_cost == 0.6]
## W = 54802, p-value = 0.8919
## alternative hypothesis: true location shift is not equal to 0
Only looking at the first half of periods
Here we have a log-log model regressing prices on shopping costs, with player-number fixed effects.
\(ln(P_{ip}) = \beta_0 + \beta_1 \delta_{i} + \beta_2 ln(S_{ip}) + \beta_3 Period_p + \epsilon_{(ip)}\)
##
## Call:
## lm(formula = log(price) ~ playerNum + log(player.transport_cost) +
## player.period_number, data = df %>% mutate(price = price +
## 0.01))
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.66267 -0.22208 0.01354 0.24857 1.05850
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.272706 0.025519 -49.873 < 2e-16 ***
## playerNumTwo Player 0.540056 0.017600 30.685 < 2e-16 ***
## log(player.transport_cost) -0.074936 0.012182 -6.151 9.45e-10 ***
## player.period_number -0.003878 0.002013 -1.926 0.0542 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3658 on 1796 degrees of freedom
## Multiple R-squared: 0.3532, Adjusted R-squared: 0.3521
## F-statistic: 326.9 on 3 and 1796 DF, p-value: < 2.2e-16
In this specification, the coefficient \(\beta_2\) measures the average effect of being assigned to the less competitive two-player treatment group. With \(\beta_2 = -0.056040\), a 1% increase in shopping costs leads to a -5.6% decrease in prices. This is significant.
Hypothesis 3. Mark-ups will be less responsive to changes in shopping costs in less competitive (lower N) markets.
\(ln(Price_{(i,p)}) = \beta_0 + \beta_1 \delta_{2p} + \beta_2 ln(ShoppingCost) + \beta_3 \delta_{i} ln(ShoppingCost) + \epsilon_{(i,p)}\)
##
## Call:
## lm(formula = log(price) ~ playerNum + log(player.transport_cost) +
## playerNum:log(player.transport_cost), data = df %>% mutate(price = price +
## 0.01))
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.68752 -0.22339 0.01848 0.25154 1.05189
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) -1.26325 0.02628 -48.071
## playerNumTwo Player 0.45058 0.04155 10.844
## log(player.transport_cost) -0.04844 0.01558 -3.108
## playerNumTwo Player:log(player.transport_cost) -0.05857 0.02464 -2.377
## Pr(>|t|)
## (Intercept) < 2e-16 ***
## playerNumTwo Player < 2e-16 ***
## log(player.transport_cost) 0.00191 **
## playerNumTwo Player:log(player.transport_cost) 0.01756 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3656 on 1796 degrees of freedom
## Multiple R-squared: 0.3539, Adjusted R-squared: 0.3528
## F-statistic: 327.9 on 3 and 1796 DF, p-value: < 2.2e-16
The coefficient \(\beta_2\) estimates that a 1% increase in shopping costs will leave to a 3.4% decrease in prices in the four-player game. The \(\beta_3\) coefficient indicates a one unit increase in shopping cost leads to a 5.9% decrease in prices in the two-player game relative to the 4-player game“.
| Dependent Var: \(ln(P_{ip})\) | Model 1 | Model 2 | ||
|---|---|---|---|---|
| \(\delta_{i}\) (two-player) | 0.559101 | *** | 0.45058 | *** |
| (0.015870) | (0.04155) | |||
| \(ln(ShoppingCost)\) | -0.056040 | *** | -0.04844 | *** |
| (0.011023) | (0.01558) | |||
| \(\delta_{i} \cdot ln(ShoppingCost)\) | -0.05857 | * | ||
| (0.02464) | ||||
| ————————————— | —— | — | —— | — |
| N | 552 | 552 |
Hypothesis 4. Collusion will be easier to form in low shopping cost environments
Define collusion
A subject is said to be ‘colluding’ when they and their adjacent players have jointly positive profits. - In the save of the two-player game, both players’ profits are positive. In the case of the four-player game, the profits of the two players to the left and right (circle marketplace) are positive. - This poses of problem in comparing “collusion” between two and four-player games. So we should not do that. - Look at violines for bit - bi-modal splits in distribution.
| Shopping Cost | Percent of Period Joint Positive Profits | Period Group Obvservation | |
|---|---|---|---|
| Two Player | 0.10 | 0.3442073 | 112 |
| Two Player | 0.25 | 0.6750000 | 56 |
| Two Player | 0.40 | 0.7213235 | 112 |
| Two Player | 0.60 | 0.8921875 | 56 |
| Four Player | 0.10 | 0.2357724 | 84 |
| Four Player | 0.25 | 0.2904762 | 42 |
| Four Player | 0.40 | 0.3745098 | 84 |
| Four Player | 0.60 | 0.4791667 | 42 |
There is visually suggestive evidence that with higher shopping costs, groups are better able to collude.
Are profits higher? - Perhaps too linked to the discussion in Hypothesis 1-3.
Tailing thing;
Compiled by Curtis Kephart, curtis.kephart@nyu.edu, with R Markdown Notebook.
2018-03-04 17:21:08 GMT, Europe/Berlin